Thus, an algorithm can be considered to be any sequence of operations that can be simulated by a Turing-complete system.

Thus for a limit ordinal, there exists a δ-indexed strictly increasing sequence with limit α.

Thus, knowing the students ' level on a developmental sequence provides information on the kind and level of knowledge they can assimilate, which, in turn, can be used as a frame for organizing the subject matter to be taught at different school grades.

Thus, the algorithm can be written as a sequence of equations

Thus the DTFT of the s sequence is also the Fourier transform of the modulated Dirac comb function .< ref group =" note "> We may also note that:

Thus we have the important result that when a discrete data sequence, s, is proportional to samples of an underlying continuous function, s ( t ), one can deduce something about the continuous Fourier transform, S ( ƒ ).

) Thus the main sequence represents the primary hydrogen-burning stage of a star's lifetime.

( Thus, they are called " macros " because a big block of code can be expanded from a small sequence of characters ).

Professional dance teachers in the US saw this approach to dancing as " extreme ", " undisciplined ", and thus, deemed it necessary to standardize the dance to present it as a sell-able commodity for the social or ballroom market Thus, compared to the mambo in Cuba, mambo had a different movement sequence in the United States and elsewhere, as it was popularized internationally.

Thus a point y in V is a cluster point of the net if and only if every neighborhood Y of y contains infinitely many elements of the sequence.

Thus represents a sequence of three phonemes,, ( the word push in standard English ), while represents the phonetic sequence of sounds ( aspirated " p "),, ( the usual pronunciation of push ).

Thus, Y and Ba are stacked in the sequence along the c-axis.

e. g. The square-free number 42 has factorisation 2 × 3 × 7, or as an infinite product: 2 < sup > 1 </ sup > · 3 < sup > 1 </ sup > · 5 < sup > 0 </ sup > · 7 < sup > 1 </ sup > · 11 < sup > 0 </ sup > · 13 < sup > 0 </ sup > · ...; Thus the number 42 may be encoded as the binary sequence < tt >... 001011 </ tt > or 11 decimal.

Thus, if in the defining sequence of a recursive function the functions and appear, then terms of the form ' g ( 5 )= 7 ' or ' h ( 3, 2 )= 10 ' might appear.

Thus, two parties talking via satellite are subject to about a half second delay in a round-trip message / response sequence.

" Thus epigenetic can be used to describe anything other than DNA sequence that influences the development of an organism.

Thus, is a bounded monotonic sequence ( in m ), which must converge.

Thus, Richard says, the real number r will not be included as any r < sub > n </ sub >, because the definition of r does not meet the criteria for being included in the sequence of definitions used to construct the sequence r < sub > n </ sub >.

Thus a valid sequence of die throws can be described by an

Thus long ē ( eta: η ) can be understood as a sequence of two short vowels: ee.

Thus, the sampled function is given by the sequence:

Thus, staining the flies for LacZ expression or activity and cloning the sequence surrounding the integration site can identify the enhancer sequence.

Thus even if the macro "" is not defined to signify non-ANSI compliance, "" will work as shown.

Thus using as defined above a Scheme implementation would rewrite "" as "", which reduces implementation's task to that of coding procedure instantiations.

Thus, Joseph Haydn remarked to his biographer Griesinger that "" He patronized me occasionally with several ducats.

Thus, there is freshness not only in the individual movements of the dance but in the shape of their continuity as well.

Thus jazz is transmuted into something holy, the sacred road to integration of being.

Thus the fictional detective is much more than a simple businessman.

Thus the cocktail party would appear to be the ideal system, but there is one weakness.

Thus in both types attention is focused on the community itself, and its phenomenological life.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

Thus human perception and human volition is the immanent cause of all social change and this most truly when the change reaches the civilizational level.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Thus, in no ordinary sense of ' simplicity ' is the Ptolemaic theory simpler than the Copernican.

Thus Burns's `` My love is like a red, red rose '' and Hopkins' `` The thunder-purple sea-beach, plumed purple of Thunder '' although clearly intelligible in content, hardly present ideas of the sort with which we are here concerned.

Thus science is the savior of mankind, and in this respect Childhood's End only blueprints in greater detail the vision of the future which, though not always so directly stated, has nevertheless been present in the minds of most science-fiction writers.

Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

Thus there is a clearer division of authority, administrative and legislative.

Thus, as a development program is being launched, commitments and obligations must be entered into in a given year which may exceed by twofold or threefold the expenditures to be made in that year.

Thus, the need for the B-70 as a strategic weapon system is doubtful.

Thus technical efficiency is achieved at the expense of actual experience.

Thus, there is an added incentive to stay on the job.

Thus, the Span of its ossification was shortened and the center's ability to `` catch up '' in ossification is demonstrated.

Thus T is not diagonalizable.

Thus Af is divisible by the minimal polynomial P of T, i.e., Af divides Af.

Thus, the study of the solutions to the equation Af is reduced to the study of the space of solutions of a differential equation of the form Af.

Thus in the three-dice example Af, Af, and the independence assumption imply that the probability that the three dice fall ace, not-ace, ace in that order is Af.

Thus we do not score the number of bull's-eyes, and the random variable is not the number of successes.

Thus if E is sufficiently small, there can be only one intersection of C and Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.